Digital Signal Processing

Modal analysis and testing

S. Ziaei Rad

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Fourier Analysis

Fourier seriesFourier TransformDiscrete Fourier series

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Fourier series

Assume that x(t) is a periodic function in time.

)sin()cos(2

)(1

0 tbtaa

tx nnn

nn

T

nn

 2

T

nn

T

nn

T

dtttxT

b

dtttxT

a

dttxT

a

0

0

0

0

) sin()(2

) cos()(2

)(2

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Fourier series(Alternative form)

1

0 )cos()(n

nnn tcctx

a

bbac

n

nnnnn )(tan    ;      122

A-

B- dtetxT

XeXtx ti

T

nn

ti

nnn

0

)(1

    )(

2|;   |

2);  Im(

2);  Re(* n

nn

nn

nnn

cX

bX

aXXX

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Fourier TransformA non-periodic function x(t) which satisfies the condition:

dttx )(

Can be represented by:

dttBtAtx ))sin()()cos()(()(

where

dtttxBdtttxA )sin()(1

)(;  )cos()(1

)(

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Fourier Transform(Alternative form)

dtetxX

deXtx

ti

ti

)(2

1)(

)()(

And

)()(2

)())(Im(

2

)())( Re(

*

XX

BX

AX

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Discrete Fourier Transform (DFT)

A function which is defined only at N discrete points can be represented by a finite series. Nktt k ,,1      

2/)1  (  2/

1

0 2sin

2cos 

2)(

NorN

nnnkk N

nkb

N

nka

axtx

where

N

k

N

kknkn

N

kk N

nkx

Nb

N

nkx

Nax

Na

1 110

2sin

1;    

2cos

1;   

2

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Discrete Fourier Transform Alternative form

1

0

/2)(N

n

Ninknkk eXxtx

NnexN

XN

k

Ninkkn ,,1       ,

1

1

/2

Note that:*rrN XX

-This form of DFT is most commonly used on digital spectrum analyser.-The DFT assumes that the function x(t) is periodic.-It is important to realize that in the DFT, there are just a discrete number of items of data in either form, i.e. N values for x and N values for Fourier series.

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DFTExample

Let N=10In time domain we have: 1021 ,,, xxx

In frequency domain, we have:

4321543210 ,,,,,,,,, bbbbaaaaaaor

)Re(),Im(),Re(),Im(),Re(

)Im(),Re(),Im(),Re(),Re(

54433

22110

alXXXXX

XXXXalX

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Discrete Fourier Transform (DFT)

x(t) x ( x(t )) k 1,N

t kT /Nk k

k

x(t) IS PERIODIC IN (FINITE) TIME T ALSO IS DEFINED

ONLY AT N DISCRETE POINTS t T / N

T

x(t)

x = x(t ) a ( a cos b sin )

a x cos

k k12 0 n

2 ntT

n 1n

2 ntT

n kk 1

2 ntT

k k

k

/

2

2WHERE Etc.

*

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DFT Spectrum

( )

( )

SAMPLING

S

MAX FREQ. :

FREQ. SPACING:

/

/

2

12

, )X( 2N

n THESE OFONLY AT AND VALUES,DISCRETE ATDEFINEDONLY ww

X( )

12 N /2

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DFT Spectrum

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DFT Spectrum

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DFT1- The input signal is digitized by an A-D converter.2- The input is recorded as a set of N discrete values, evenly spaced in period T.3-The sample is periodic in time T.4- There is a relation between the sample length T, the number of discrete values N, the sampling (or digitizing) rate and the range of resolution of the frequency spectrum, i.e. .

is called the Nyquist frequency.

s

,   max

s

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DFT

TN

T

N

s

s

2

)   2

(2

1

2max

1- Usually, the size of transform (N) is fixed for an analyser, therefore the frequency range and frequency resolution is only determined by the length of the sample.2- The basic equation (*) will be used to determine the coefficient

,,,,,, 21210 bbaaa

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DFTThe basic equation that is solved to determine spectral composition is:

1

1

0

3

2

1

)/2cos(5.0

)/6cos(5.0

)/4cos(5.0

)/2cos(5.0

b

a

a

TN

T

T

T

x

x

x

x

N

or

}{][}{}]{[}{ 1

knnk xCaaCx

**

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FFT-Some efforts has been devoted to equation (**) for calculation of spectral coefficients.-Cooley and Tukey (1960) introduced an algorithm called “Fast Fourier Transform (FFT)”. The method requires N to be an integral power of 2 and the values usually taken is between 256 to 4096.-There are number of features of digital Fourier analysis, if not properly treated, can give rise to erroneous results. These are generally the result of discretisation approximation and the limited length of the time history. - Aliasing - Zooming - Leakage - Averaging - Windowing - Filtering

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Aliasing- Aliasing is a problem from discretisation of the originally continuous time history. - With this descretisation process, the existence of very high frequency in the original signal may well be misinterpreted if the sampling rate is too slow.

The phenomenon of aliasing a- Low-frequency signal b- High-frequency signal

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Leakage

- Leakage is a direct consequence of a finite length of time history.-In Fig. a, the signal is perfectly periodic in the time window T and the spectrum is a single line.-In Fig. b, the periodicity assumption is not valid and the spectrum is not at a single frequency.-Energy has ‘leaked’ into a number of spectral lines in close to the true frequency.-Leakage is a serious problem in many application of DSP, including FRF measurements.

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Leakage

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LeakageWays of avoiding leakage:-Changing the duration of the measurement sample length to match any underlying periodicity in the signal. However, it is difficult to determine the period of signal.-Increasing the duration of the measurement period, T, so that the separation between spectral lines is finer.-Adding zeros to the end of the measured sample (zero padding) In this way we partially achieving the preceding result but without requiring more data.-Or by modifying the signal sample obtained in such a way to reduce the severity of leakage. The process is called ‘windowing’.

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Windowing- In many situation, the most practical solution to the leakage problem is windowing.- There are a range of different windows for different classes of problems.-Windowing is applied to the time signal before performing the Fourier Transform. x(t) measured signal w(t) window profileOr in Frequency Domain

Where * denotes the convolution process.

)()()(' twtxtx

)(*)()(' wXX

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Windowing

Boxcar

ExponentialHanning

Cosine-taper

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WindowingBoxcar

Hanning

Cosine-taper

Exponential

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Windowing

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Effect of Hanning Window on Discrete

Fourier Transform

))/2cos(1(5.)( Tttw

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Filtering

-This is another signal conditioning process which has a direct parallel with windowing.-In filtering, we simply multiply the original signal spectrum by the frequency characteristic of the filter. or in time domain -Common types of filters are:

- High pass-Low pass-Narrow-band-Notch

)()()(' WXX )(*)()(' twtxtx

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Filtering

High-pass

Band-limited

Narrow-band

Notch

Frequency and time domain characteristics of common filters

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Improving resolutionInadequate frequency resolution especially at lower end of the frequency range and for lightly-damped systems occurs because of -limited number of discrete points available - maximum frequency range to be covered - the length of time sample

Possible actions to improve the resolution-Increasing transform size-Zero padding-Zoom

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Increasing transform size

-An immediate solution to this problem would be to use larger transform. This gives finer resolution around the region of interest. This caries the penalty of providing more information than required.-Until recently, the time and storage requirements to perform the DFT were a limiting factor.- Transform size of order 2000 to 8000 are standard.

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Zero padding

- To maintain the same overall frequency range, but to increase resolution by n, a signal sample of n times the duration is needed.-One way is to add a series of zeros to the short sample of actual signal to create a new sample which is longer than the original measurement and thus provide the desire finer resolution.-The fact is that no additional have been provided while apparently greater detail in the spectrum is achieved. It is not a genuinely finer spectrum, rather it is the coarser spectrum that interpolated and smoothed by the extension of the analysed record.-An example of the effect and potential dangers of zero padding is shown in the next slide.

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Zero padding

Results using zero padding toimprove resolution.a- DFT of data between 0 to T1b- DFT of data padded to T2c- DFT of full record 0 to T2

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Zoom- The common solution to the need for finer frequency resolution is to zoom on the frequency range of interest and to concentrate all the lines into a narrow band between .-There are different ways of doing this but perhaps the easiest one is to use a frequency shifting process coupled with a controlled aliasing device.

maxmin      ftof

Spectrum of signal

Band-pass filter

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Averaging

-When analyzing random vibrations signal, we obtain estimates for the spectral densities and correlation functions which are used to characterize this type of signal. -Generally, it is necessary to perform an averaging process, involving several individual time records, before a confident results is obtained.-Two major considerations which determine the number of average: - the statistical reliability - the removal of random noise from the signals

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Different interpretations of multi-sample averaging

Sequential

Overlap